Hermitian Jacobi Forms of Higher Degree

Abstract

We develop the theory of Hermitian Jacobi forms in degree $n > 1$. This builds on the work of Klaus Haverkamp in \cite{HThesis} who developed this theory in degree $n = 1$. Haverkamp in turn generalized a monograph of Eichler and Zagier, \cite{E&Z}. Hermitian Jacobi forms are holomorphic functions which appear in certain infinite series expansions (Fourier Jacobi expansions) of Hermitian modular forms. In this work we give a definition of Hermitian Jacobi forms in degree $n > 1$, give their relationship to more classical Hermitian modular forms and construct a useful tool for studying Hermitian Jacobi forms, the theta expansion. This theta expansion allows us to relate our forms to classical modular forms via the Eichler-Zagier map and thereby bound the dimension of our space of forms. We then go on to apply the developed theory to prove some non-vanishing results on the Fourier coefficients of Hermitian modular forms.